Why use f(x)?

Date: 02/14/99 at 22:27:02
From: ed brown
Subject: Why use f(x)?
Can you explain the point of using f(x) in an equation as opposed to y?
Thanks in advance.

Date: 02/15/99 at 08:42:50
From: Doctor Jerry
Subject: Re: Why use f(x)?
One can use the notation y = x^2-5x+6 for many purposes. The equation
can carry the idea that as x changes, y changes. It doesn't give,
explicitly, the name of the rule that says that to the number x
corresponds the number x^2-5x+6.
When we write y = sin(x), we are being redundant, for some purposes.
"sin" is the name of a particular rule. Out of habit, we may say "graph
the equation y = sin(x)." We could say: graph the function sin.
I expect that you also use log or ln, tan, cos, and maybe sqrt(x).
Mathematicians often like to be minimalists. If they have a certain
rule of correspondence in mind, they may prefer naming it by a letter,
such as f. They don't need y = f(x); just f will do.
When we want to put two functions together, like ln(sin(x)), we can
name the new function as ln o sin. Of course, this can also be done, as
you may prefer, by saying something like
Y = ln(w)
w = sin(x),
so, y = ln(sin(x)).
Another reason is that calculus is the study of functions, not
equations. We study functions f and their derivatives f'. We integrate
functions. All of this is possible with the y = x^2+5x+6 notation, but
somewhat more awkward, and it takes more talking to be precise.
Finally, if f(x) = x^2 and we want to calculate (as is needed in
calculus) the difference quotient (this is the slope of the line
joining (x,f(x)) and (x+h,f(x+h)))
[f(x+h)-f(x)]/h,
how could you easily give the recipe using the y = x^2 notation?
- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/